Wednesday, June 18, 2008

Replacement players, VORP, salaries, and MRP

In a post on his blog, J.C. Bradbury argues, again, that a player's free-agent player salary is equal to his "MRP," which means "marginal value of production." That is, a player should be paid exactly the amount by which his performance increases the team's revenue.

And he seems to think that revenue is exactly proportional to the player's performance, rather than the player's performance as measured against replacement level. Because of that, he dismisses the concept of replacement value (and VORP). But I don't understand why he would do that.

The idea behind MRP is this: the more employees you hire, the less each one contributes to the bottom line. If you're running a Wal-Mart, you might want ten cashiers. If you hire an eleventh cashier, it might help a little bit: if there's a crowd of customers, fewer might leave the store if the lineups are shorter. But the eleventh is only useful in busy times, so he's worth less than the other 10.

The idea is this: suppose cashiers earn $30,000 a year. The first five or six might bring the company $70,000 in revenue each. The seventh might bring in only $60K. The eighth adds $50K, the ninth $40K, the tenth $30K, and the eleventh $20K. The eleventh cashier is actually losing the company money, so she never gets hired in the first place. And the last cashier hired brought in $30,000, which exactly matches his salary. Thus the equivalence: salary = MRP.

That works for Wal-Mart, but not for baseball. Why? Because in baseball, the number of employees is fixed, and so is the minimum salary. At Wal-Mart, if you have 11 cashiers, you figure that the 11th costs $30,000 but is bringing in only $20,000 in revenue. So you fire him. In baseball, you might figure that you're paying the 25th man $390,000, but he's contributing nothing to the bottom line (because he gets no playing time). So you want to release him. But you can't – there's a rule requiring you to have 25 men on your roster. And there's a minimum salary of $390,000. So you're stuck. In this case, the MRP of the 25th man is less than his salary.

It can work the other way around, too. Suppose you're a big-market team with lots of fans who love to win, and you figure that a 25th man, while costing only $390,000, is bringing in revenues of over a million. You'd like to hire a 26th player, who would bring in another $900,000 or so. But, unlike Wal-Mart, you can't go hiring that extra player. There are rules against that. In that case, the 25th man is earning less than his MRP.

So, in baseball, a player's salary could easily be more, or less, than his MRP.

The real-world equivalence between salary and MRP is

Salary = MRP

But that's a special case that just happens to apply to Wal-Mart. I would argue that the more general equivalence is

Salary over and above the alternative = MRP over and above the alternative

At Wal-Mart, the alternative is "nobody" – you just never hire the 11th cashier. That alternative has zero salary and zero MRP, so the second equation collapses into the first equation. But in baseball, the alternative is NEVER "nobody" – you have to fill the roster spot, whether you want to or not. The alternative is a player at minimum salary, creating a replacement-level MRP. It's one of the many freely available minor-leaguers. That means

Salary over and above the $390,000 minimum = MRP over replacement player

If you choose to define "VORP" in terms of dollars instead of runs, you get

Salary - $390,000 = VORP

Which, I think, is what's really happening in baseball. J.C. doesn't agree with that formulation – he wants to stick with "salary = MRP". He wants to value marginal runs from zero, rather than from replacement value. But, I argue, that clearly leads to untenable conclusions.

For instance, suppose a marginal win is worth $5 million. Then a marginal run is worth about $500,000.

Suppose a replacement-level player creates 39 runs. At $50,000 per run, you'd expect him to cost $19.5 million. But you can pick up any one of these guys for $390,000! So "salary = MRP" just doesn't make sense.

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If you don't buy that argument, here's another one. Suppose you really, truly believe that a player earns his MRP. And suppose the 25th guy on your roster earns $390,000, the MLB minimum.

Now, halfway through the season, the union and MLB agree to double the minimum salary. The 25th guy gets to keep his job – after all, the team has to have 25 guys, and this is still the best one available. But now he's making $780,000.

His salary doubled, but his MRP, obviously, is exactly the same! So even if his salary was equal to MRP before, it certainly isn't now. Which means that there's no reason to have expected them to be equal in the first place.

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One of J.C.'s arguments is that not all replacement-level players are worth only $390,000. It could be that all the players eligible for the minimum are young draft choices, and you don't want to use up their "slave" years if your season is a lost cause – you'd rather save them for when your team is a contender. In that case, you might have to sign a replacement-level veteran for $1,000,000 or so.

To which I say: there is no shortage of mediocre veterans who can be had for $390K, that you would have to spend a million. If you DO spend a million, it's probably because you peg the veteran as better than replacement. The extra $610,000 is worth it if you expect about 12 1.2 runs better than replacement over a full season, which isn't a lot.

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By the way, you could argue that a player is never paid less than his MRP. In a sense, even if the 25th guy on your roster never bats, his presence contributes more than $390,000. That's because if you released him, and didn't call anyone up, the commissioner would fine you a lot more than $390,000.

But that's a trick technicality, and it's not what J.C. is arguing here.

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Finally, I am puzzled by J.C.'s dislike of VORP because, according to him, it's an insider term and hard to explain:

The big advantage of these is that I can have these conversations with people other than die-hard stat-heads ... I view VORP as an insider language, and by using it you can signal that you are insider. It’s like speaking Klingon at a Star Trek convention. I can signal to others who speak the language that I am one of you. But, the danger of VORP is that once you bring it up the discussion goes down the wrong path as the uninitiated have reason to feel they are being told they are not as smart as the person making the argument. It’s like constantly bringing up the fact that you only listen to NPR or watch the BBC news at dinner parties. The response is likely going to be the same, “well fuck you too, you pretentious asshole!”

But, as I think a commenter on J.C's site points out, "MRP" is also pretty jargon-y insider economist talk, isn't it? And it's a lot hard to explain than VORP. So I'm a bit confused by J.C.'s aversion to the term. How come sabermetric abbreviations are pompous, but economics abbreviations are not?

I'm arguing that to get his free-agent salary, you don't take his MRP. You take his MRP, subtract the replacement player's MRP. You take that add $390,000, and you get his MRP.

Suppose the value of a run is $100K. J.C. would argue that if a player creates 40 runs, he's worth $4 million. I argue that if a replacement player creates 39 runs, the guy who creates 40 is worth $390,000 for the first 39 runs, plus $100,000 for the extra run -- $490,000 total.

The example given in the comment makes more sense then those given in the post. The examples you gave in the post all have to do with players that are not yet free agents. Teams have monopoly power at that point, so no one is going to get paid anything near what they are worth. That being said, I did like the point you made in the response.

Wait I retract my last comment. Your whole thinking is skewed because, fundamentally, those replacement runs are not worth 490K, they are only that much because owners have a monopoly on players before they reach free agency. On free agent market, players are actually paid what they are worth, so Bradbury is right. You can't compare a free agent to a AAA call up because no one can bid on what to pay the call up. Value above replacement can be used to figure out salaries, but not the way you are arguing for here.

"On the free agent market players are actually paid what they are worth". That''s actually a problematic statement if by "worth" you mean something that can in theory be translated into a particular and objective run or win value. Player X "worth" 40 runs on the baseball field may be worth more in dollars to a large market team than a small market team, because the extra wins generated by 40 runs added to a large market team translates to more extra TV viewers etc. So the Yankees can bid $10 million for Player X and his 40 runs and the Royals only $3 million and his 40 runs. He is "worth" $10 million to the Yanks and "worth" $3 million to the Royals, but he's the same player on the field, generating the same 40 runs regardless.

Given that, it's hard to argue that one can assign a set objective value to a run generated by a free agent. Stats like VORP try to evaluate a player's value within the confines of the playing field -- how valuable the player is in helping his team win a baseball game, which is essentially the same from team to team as the playing rules are the same for all. That's a different issue than free agent salary value, which varies from team to team as each team is playing by different revenue-generating rules.

JC stretches the definition of MRP to a statement about average values; this can never work. And you are right to doubt HIS interpretation of MRP (which is incorrect).

But VORP fails operationally. You suggest that I need only offer $1 million - $390,000 = $610,000 to a player that can generate $1 million for my team. Suppose another owner and the player can generate $610,001 for them. Sounds like I would lose the player to the another owner who bids according to MRP unless I bid closer to his MRP for me.

Now, MRP is continuous theoretically and real markets are more lumpy. So, we expect pay (if competitively determined) to be between the highest and second highest value in the league. And those values are determined by the MRP in each location.

Except for players at the minimum. But then wouldn't we expect the league to choose the minimum so that the roster limit is met by hiring those few last players at the minimum by those teams that only value those players at the minimum amount? MRP works again.

"You suggest that I need only offer $1 million - $390,000 = $610,000 to a player that can generate $1 million for my team. Suppose another owner and the player can generate $610,001 for them. Sounds like I would lose the player to the another owner who bids according to MRP unless I bid closer to his MRP for me."

No, you wouldn't. Because the other owner wouldn't bid $610,000 if the player only generated $610,001, because its alternative is a player who would generate (say) $500,000 for only $390,000.

Remember, you only get 25 players. There is a "gift" by which, if you take the marginal player, who is just barely out of a job, you get (say) $500K worth of MRP for $390,000. And that profit won't be competed away *because all teams only get 25 players*. No team can just keep buying these players for $390K and profiting the proceeds (until the MRP of the 97th player (or whatever) drops to $390K.

You like corn flakes. The first box is worth $10 to you, the second box is worth $5 to you. Corn Flakes sell for $4.99. You'll buy two boxes. But now, suppose Bud Selig tells you, if you don't buy a second box, we'll force you to buy a half a second box for $1. Now will you buy the second box? No. The marginal benefit over the half-box is now $4, but the cost is $4.99.

What I'm saying is: consider $390K a fixed cost, because you have to pay that no matter what. And consider (say) 24 runs a fixed benefit, because you can get that no matter who you hire at $390K. Now, if you consider the MRP the marginal revenue product *over the fixed benefit*, and you consider the cost *the salary minus the fixed cost*, only THEN does salary=MRP.

Another way to say it: "M" stands for "marginal". But not by adding a brand new player at the margin, but by *replacing the default player* at the margin.

But reductio ad absurdem, nobody would ever have to pay more than $309,000 plus $1 (or two or three) for ANY player by this VORP logic.

And that clearly cannot stand under competition over inputs.

You are correct that you replace the default player at the margin, but the value of the player at their second highest use is least any player can make.

You seem hung up on the minimum salary requirement, but this just means that the last person hired will only be worth $309,000, and that will only happen on teams where the last player hired is worth this amount. Not all teams have minimum salary players.

How much will the best player earn? You say $800K. Suppose team A pays him that, and gets $800K worth of production.

How much will the next best player earn? You say team B pays him $700K, and gets $700K of production.

How much will the third player earn? $390K, because only one team is bidding. But he earns team C $500K in production. He earns less than MRP, giving team C a profit of $110K.

If teams A and B are rational, they will never bid MRP for the better players unless they can get them for at least $110K less than the value of their production! Because they can always revert to being like team C and get their $110K that way.

Mathematically, they will be willing to pay $390K for the player's first 50 runs, *and MRP after that*. So the guy that creates 80 runs gets $390 for his first 50, and $300K for his next 30 (MRP), for a total of $690.

A nicely structured example with built-in truncation still misses the point.

You admit that the first two go for their MRP. And you are simply incorrect about the remaining players. Why do they suddenly turn into $390,000 players when they clearly are worth more? One of the other teams will bid them up to their contribution price. And that's MRP again.

The only players this will not matter about will be players worth less than $390,000. They will never be employed.

And a league acting in its own interest will never choose a minimum where they have to pay more than the bottom-margin players are worth.

And remember, we're talking in a world of certainty. In a world of uncertainty, it'll all be expected MRP. Occasionally, there will be bargains and busts but, on average, any bias one way or the other again opens profit potential taking advantage of the bias.